💡Alright, so I have this determinant to evaluate. It's a 3x3 matrix, which I remember involves some expansion, maybe using minors or cofactors. Let me write it down first to visualize it better:[begin{vmatrix} 1 & x & z 1 & x + z & 2z 1 & x & x + 2z end{vmatrix}]Hmm, okay. Since it's a 3x3 determinant, I think the standard method is to expand along the first row. I remember something about minors and signs alternating. Let me recall the formula for expanding a determinant along the first row:The determinant is equal to the sum of each element in the first row multiplied by its corresponding cofactor. The cofactor involves the minor (which is the determinant of the submatrix) and a sign based on the position, which is (-1)^(i+j) where i is the row and j is the column.So, for the first row, the elements are 1, x, and z. Their positions are (1,1), (1,2), and (1,3). So the signs will be (+), (-), and (+) respectively because (-1)^(1+1)=1, (-1)^(1+2)=-1, and (-1)^(1+3)=1.So, breaking it down:1 * det(minor of 1) - x * det(minor of x) + z * det(minor of z)Now, let's find each minor. The minor of an element is the determinant of the submatrix that remains after deleting the row and column containing that element.Starting with the first element, 1 at (1,1). If I remove the first row and first column, the submatrix is:[begin{vmatrix} x + z & 2z x & x + 2z end{vmatrix}]So, the determinant of this 2x2 matrix is (x+z)(x+2z) - (2z)(x). Let me compute that:(x+z)(x+2z) = x^2 + 2xz + xz + 2z^2 = x^2 + 3xz + 2z^2Then subtract 2zx: x^2 + 3xz + 2z^2 - 2xz = x^2 + xz + 2z^2Okay, so the minor for the first element is x^2 + xz + 2z^2.Next, the minor for the second element, x at (1,2). Removing the first row and second column, the submatrix is:[begin{vmatrix} 1 & 2z 1 & x + 2z end{vmatrix}]The determinant here is (1)(x + 2z) - (2z)(1) = x + 2z - 2z = xSo, the minor for the second element is x.Wait, hold on, that seems too simple. Let me double-check:First element of the submatrix: 1 * (x + 2z) = x + 2zSecond element: 2z * 1 = 2zSo, determinant is (x + 2z) - 2z = x. Yeah, that's correct.Now, the minor for the third element, z at (1,3). Removing the first row and third column, the submatrix is:[begin{vmatrix} 1 & x + z 1 & x end{vmatrix}]The determinant here is (1)(x) - (x + z)(1) = x - x - z = -zSo, the minor for the third element is -z.Putting it all together, the determinant is:1*(x^2 + xz + 2z^2) - x*(x) + z*(-z)Wait, no, hold on. The cofactors have signs based on their position. So, the first term is positive, the second is negative, and the third is positive. So, it's:1*(x^2 + xz + 2z^2) - x*(x) + z*(-z)Wait, no, actually, the cofactor for the third element is z multiplied by the minor, which is -z, but also multiplied by the sign from the cofactor expansion. Wait, I think I might be confusing the signs.Let me clarify. The expansion is:1 * [det(minor1)] - x * [det(minor2)] + z * [det(minor3)]So, the signs are already accounted for in the expansion. So, the first term is positive, the second is subtracted, and the third is added.So, substituting the minors:1*(x^2 + xz + 2z^2) - x*(x) + z*(-z)Compute each term:First term: x^2 + xz + 2z^2Second term: -x^2Third term: -z^2So, combining all terms:x^2 + xz + 2z^2 - x^2 - z^2Simplify:x^2 - x^2 cancels out.xz remains.2z^2 - z^2 = z^2So, total determinant is xz + z^2Factor out z:z(x + z)Wait, but in the initial calculation, I thought the answer was z(3x + z). Hmm, that's conflicting.Wait, maybe I made a mistake in computing the minors. Let me double-check each minor.First minor: for element 1 at (1,1), submatrix is:[begin{vmatrix} x + z & 2z x & x + 2z end{vmatrix}]Compute determinant:(x + z)(x + 2z) - (2z)(x) = x^2 + 2xz + xz + 2z^2 - 2xz = x^2 + xz + 2z^2That seems correct.Second minor: for x at (1,2), submatrix is:[begin{vmatrix} 1 & 2z 1 & x + 2z end{vmatrix}]Determinant: (1)(x + 2z) - (2z)(1) = x + 2z - 2z = xThat's correct.Third minor: for z at (1,3), submatrix is:[begin{vmatrix} 1 & x + z 1 & x end{vmatrix}]Determinant: (1)(x) - (x + z)(1) = x - x - z = -zThat's correct.So, substituting back:1*(x^2 + xz + 2z^2) - x*(x) + z*(-z) = x^2 + xz + 2z^2 - x^2 - z^2 = xz + z^2So, the determinant is xz + z^2, which factors to z(x + z)But wait, in the initial thought process, I thought the answer was z(3x + z). That must have been a mistake.Wait, maybe I should try expanding along a different row or column to see if I get the same result.Alternatively, perhaps I made a mistake in the expansion.Let me try expanding along the first column instead, since all the elements are 1s, which might simplify things.The determinant can be expanded along the first column as well. The formula is similar, but now the signs will be based on the column index.So, expanding along the first column:1 * det(minor1) - 1 * det(minor2) + 1 * det(minor3)Where minor1 is the submatrix removing first row and first column:[begin{vmatrix} x + z & 2z x & x + 2z end{vmatrix}]Which we already computed as x^2 + xz + 2z^2Minor2 is the submatrix removing second row and first column:[begin{vmatrix} x & z x & x + 2z end{vmatrix}]Determinant: x*(x + 2z) - z*x = x^2 + 2xz - xz = x^2 + xzMinor3 is the submatrix removing third row and first column:[begin{vmatrix} x & z x + z & 2z end{vmatrix}]Determinant: x*2z - z*(x + z) = 2xz - xz - z^2 = xz - z^2So, putting it all together:1*(x^2 + xz + 2z^2) - 1*(x^2 + xz) + 1*(xz - z^2)Compute each term:First term: x^2 + xz + 2z^2Second term: -x^2 - xzThird term: +xz - z^2Combine all terms:x^2 - x^2 cancels out.xz - xz + xz = xz2z^2 - z^2 = z^2So, total determinant is xz + z^2, same as before.So, that confirms it. The determinant is xz + z^2, which factors to z(x + z)But wait, in the initial problem, the assistant's answer was z(3x + z). That must have been incorrect.Wait, perhaps I made a mistake in the initial expansion? Let me check again.Wait, in the initial expansion, I think I might have miscalculated the minors or the signs.Wait, no, in the first expansion, I had:1*(x^2 + xz + 2z^2) - x*(x) + z*(-z) = x^2 + xz + 2z^2 - x^2 - z^2 = xz + z^2But in the assistant's solution, they had:((x+z)(x+2z) - 2zx) - x((x + 2z) - 2z) + z(x - (x + z)) = (x^2 + 3xz + 2z^2 - 2zx) - x^2 - z^2 = 3xz + z^2Wait, so in their calculation, they had 3xz + z^2, which factors to z(3x + z). But in my calculation, I have xz + z^2.So, there's a discrepancy here. Let me see where the mistake is.Looking back at the assistant's steps:They expanded along the first row:1 * det(minor1) - x * det(minor2) + z * det(minor3)Computed minor1 as (x+z)(x+2z) - 2zx = x^2 + 3xz + 2z^2 - 2zx = x^2 + xz + 2z^2Then minor2: (x + 2z) - 2z = xMinor3: x - (x + z) = -zSo, putting together:1*(x^2 + xz + 2z^2) - x*(x) + z*(-z) = x^2 + xz + 2z^2 - x^2 - z^2 = xz + z^2But in the assistant's solution, they had:((x+z)(x+2z) - 2zx) - x((x + 2z) - 2z) + z(x - (x + z)) = (x^2 + 3xz + 2z^2 - 2zx) - x(x + 2z - 2z) + z(x - x - z) = (x^2 + xz + 2z^2) - x^2 - z^2 = xz + z^2Wait, but in the assistant's final answer, they wrote 3xz + z^2. That must be a mistake.Wait, no, looking back, in their calculation, they had:((x+z)(x+2z) - 2zx) = x^2 + 3xz + 2z^2 - 2zx = x^2 + xz + 2z^2Then -x*((x + 2z) - 2z) = -x*(x) = -x^2Then +z*(x - (x + z)) = +z*(-z) = -z^2So, total is x^2 + xz + 2z^2 - x^2 - z^2 = xz + z^2But in their final answer, they wrote 3xz + z^2, which is incorrect.So, the correct determinant is xz + z^2, which factors to z(x + z)Therefore, the assistant made a mistake in their final answer, probably a typo or miscalculation.So, in conclusion, the determinant evaluates to z(x + z)

💡To determine the fraction of the eighth square that would be shaded, we follow these steps:1. **Identify the pattern of shaded squares**: The number of shaded squares in each subsequent diagram follows the sequence of square numbers, defined by ( n^2 ).2. **Calculate the number of shaded squares in the eighth diagram**: Using the formula for square numbers, we calculate the eighth term: [ 8^2 = 64 ]3. **Identify the pattern of total squares**: The total number of smaller squares in each diagram also follows the sequence of square numbers. For the eighth diagram, this is again: [ 8^2 = 64 ]4. **Calculate the fraction of the diagram that is shaded**: The fraction of the eighth square that is shaded is the ratio of the number of shaded squares to the total number of squares: [ frac{text{Number of shaded squares}}{text{Total number of squares}} = frac{64}{64} = 1 ]5. **Conclusion**: The fraction of the eighth square that is shaded is ( 1 ).The final answer is (boxed{E}).

💡Alright, so I have this problem here about negating a proposition. The original statement is: "For all ( x in mathbb{R} ), ( x^2 - 2x + 4 leq 0 )." I need to find its negation from the given options.First, I remember that negating a universal statement ("for all") turns it into an existential statement ("there exists"). So, the negation of "For all ( x ), P(x)" is "There exists an ( x ) such that not P(x)." Applying that here, the negation should be: "There exists an ( x in mathbb{R} ) such that ( x^2 - 2x + 4 > 0 )."Looking at the options, option C says exactly that: "There exists ( x in mathbb{R} ), ( x^2 - 2x + 4 > 0 )." So, that seems like the right answer.But wait, let me double-check. Maybe I should analyze the quadratic expression ( x^2 - 2x + 4 ) to see if it's always positive or not. If it's always positive, then the original statement is false, and the negation would be true.To analyze ( x^2 - 2x + 4 ), I can complete the square. Let's see:( x^2 - 2x + 4 = (x^2 - 2x + 1) + 3 = (x - 1)^2 + 3 ).Since ( (x - 1)^2 ) is always non-negative, adding 3 makes the entire expression always greater than or equal to 3. So, ( x^2 - 2x + 4 ) is always greater than 0 for all real numbers ( x ).That means the original statement "For all ( x in mathbb{R} ), ( x^2 - 2x + 4 leq 0 )" is false because the quadratic is never less than or equal to zero. Therefore, its negation, which is "There exists ( x in mathbb{R} ), ( x^2 - 2x + 4 > 0 )," is true.So, confirming again, option C is correct.

💡Okay, so I need to find the quotient when the polynomial (8x^4 + 4x^3 - 6x^2 + 2x - 5) is divided by (x - 1). Hmm, I remember that polynomial division is similar to long division with numbers, but I'm a bit rusty on the exact steps. Let me try to recall.First, I think I need to set up the division. The dividend is (8x^4 + 4x^3 - 6x^2 + 2x - 5), and the divisor is (x - 1). So, I'll write it like this:``` _______________________x - 1 | 8x^4 + 4x^3 - 6x^2 + 2x - 5```Now, I need to divide each term of the dividend by the leading term of the divisor. The leading term of the divisor is (x), and the leading term of the dividend is (8x^4). So, I divide (8x^4) by (x), which gives me (8x^3). That's the first term of the quotient.Next, I multiply the entire divisor (x - 1) by (8x^3) and subtract the result from the dividend. Let's do that:Multiply (8x^3) by (x - 1):(8x^3 times x = 8x^4)(8x^3 times (-1) = -8x^3)So, the product is (8x^4 - 8x^3). Now, subtract this from the original dividend:[(8x^4 + 4x^3 - 6x^2 + 2x - 5) - (8x^4 - 8x^3) = 0x^4 + (4x^3 + 8x^3) - 6x^2 + 2x - 5 = 12x^3 - 6x^2 + 2x - 5]Okay, so now the new dividend is (12x^3 - 6x^2 + 2x - 5). I repeat the process: divide the leading term (12x^3) by the leading term of the divisor (x), which gives me (12x^2). That's the next term in the quotient.Multiply (12x^2) by (x - 1):(12x^2 times x = 12x^3)(12x^2 times (-1) = -12x^2)Subtract this product from the current dividend:[(12x^3 - 6x^2 + 2x - 5) - (12x^3 - 12x^2) = 0x^3 + (-6x^2 + 12x^2) + 2x - 5 = 6x^2 + 2x - 5]Now, the new dividend is (6x^2 + 2x - 5). Divide the leading term (6x^2) by (x) to get (6x). That's the next term in the quotient.Multiply (6x) by (x - 1):(6x times x = 6x^2)(6x times (-1) = -6x)Subtract this from the current dividend:[(6x^2 + 2x - 5) - (6x^2 - 6x) = 0x^2 + (2x + 6x) - 5 = 8x - 5]Now, the new dividend is (8x - 5). Divide (8x) by (x) to get (8). That's the next term in the quotient.Multiply (8) by (x - 1):(8 times x = 8x)(8 times (-1) = -8)Subtract this from the current dividend:[(8x - 5) - (8x - 8) = 0x + (-5 + 8) = 3]So, the remainder is (3). Since the degree of the remainder (which is 0) is less than the degree of the divisor (which is 1), we stop here.Putting it all together, the quotient is (8x^3 + 12x^2 + 6x + 8) with a remainder of (3). Wait, that doesn't match the previous answer. Did I make a mistake somewhere?Let me check my steps again. When I subtracted after multiplying (6x), I had:[(6x^2 + 2x - 5) - (6x^2 - 6x) = 0x^2 + (2x + 6x) - 5 = 8x - 5]That seems correct. Then, dividing (8x) by (x) gives (8), which is correct. Multiplying (8) by (x - 1) gives (8x - 8), and subtracting that from (8x - 5) gives (3). So, the remainder is indeed (3).But in the initial answer, the remainder was (-25). Hmm, that suggests I might have made a mistake in my calculation. Let me try synthetic division as an alternative method to verify.Using synthetic division with (x = 1):The coefficients of the dividend are (8, 4, -6, 2, -5).Set up the synthetic division:1 | 8 4 -6 2 -5 8 12 6 8 ------------------------- 8 12 6 8 3So, the coefficients of the quotient are (8, 12, 6, 8) and the remainder is (3). Therefore, the quotient is (8x^3 + 12x^2 + 6x + 8) with a remainder of (3).Wait, this contradicts the initial answer. I think I must have made a mistake in the initial long division. Let me go back to the long division and see where I went wrong.Looking back at the initial long division steps:1. Divided (8x^4) by (x) to get (8x^3).2. Multiplied (8x^3) by (x - 1) to get (8x^4 - 8x^3).3. Subtracted to get (12x^3 - 6x^2 + 2x - 5).4. Divided (12x^3) by (x) to get (12x^2).5. Multiplied (12x^2) by (x - 1) to get (12x^3 - 12x^2).6. Subtracted to get (6x^2 + 2x - 5).7. Divided (6x^2) by (x) to get (6x).8. Multiplied (6x) by (x - 1) to get (6x^2 - 6x).9. Subtracted to get (8x - 5).10. Divided (8x) by (x) to get (8).11. Multiplied (8) by (x - 1) to get (8x - 8).12. Subtracted to get (3).So, according to this, the quotient is (8x^3 + 12x^2 + 6x + 8) with a remainder of (3). But the initial answer had a quotient of (8x^3 + 12x^2 - 18x + 20) and a remainder of (-25). Clearly, there's a discrepancy.Wait, perhaps the initial answer was incorrect. Let me check the synthetic division again. If I use (x = 1), the remainder should be the value of the polynomial at (x = 1). Let's compute that:(8(1)^4 + 4(1)^3 - 6(1)^2 + 2(1) - 5 = 8 + 4 - 6 + 2 - 5 = 3). So, the remainder is indeed (3), which matches my synthetic division result.Therefore, the initial answer must have been incorrect. The correct quotient should be (8x^3 + 12x^2 + 6x + 8) with a remainder of (3).Wait, but in the initial long division, the quotient was written as (8x^3 + 12x^2 - 18x + 20) with a remainder of (-25). That suggests that perhaps the initial setup was wrong or there was a miscalculation in the subtraction steps.Let me try the long division again carefully.Dividing (8x^4 + 4x^3 - 6x^2 + 2x - 5) by (x - 1):1. (8x^4 ÷ x = 8x^3).2. Multiply (8x^3) by (x - 1): (8x^4 - 8x^3).3. Subtract: ((8x^4 + 4x^3) - (8x^4 - 8x^3) = 12x^3).4. Bring down the next term: (-6x^2).5. (12x^3 ÷ x = 12x^2).6. Multiply (12x^2) by (x - 1): (12x^3 - 12x^2).7. Subtract: ((12x^3 - 6x^2) - (12x^3 - 12x^2) = 6x^2).8. Bring down the next term: (2x).9. (6x^2 ÷ x = 6x).10. Multiply (6x) by (x - 1): (6x^2 - 6x).11. Subtract: ((6x^2 + 2x) - (6x^2 - 6x) = 8x).12. Bring down the next term: (-5).13. (8x ÷ x = 8).14. Multiply (8) by (x - 1): (8x - 8).15. Subtract: ((8x - 5) - (8x - 8) = 3).So, the quotient is (8x^3 + 12x^2 + 6x + 8) with a remainder of (3). This matches the synthetic division result.Therefore, the initial answer was incorrect. The correct quotient is (8x^3 + 12x^2 + 6x + 8) with a remainder of (3).Wait, but in the initial problem, the user provided an answer with a quotient of (8x^3 + 12x^2 - 18x + 20) and a remainder of (-25). That suggests that perhaps the initial setup was wrong or there was a miscalculation in the subtraction steps.Let me check the initial long division steps again as presented:1. Divided (8x^4) by (x) to get (8x^3).2. Multiplied (8x^3) by (x - 1) to get (8x^4 - 8x^3).3. Subtracted to get (12x^3 - 6x^2 + 2x - 5).4. Divided (12x^3) by (x) to get (12x^2).5. Multiplied (12x^2) by (x - 1) to get (12x^3 - 12x^2).6. Subtracted to get (6x^2 + 2x - 5).7. Divided (6x^2) by (x) to get (6x).8. Multiplied (6x) by (x - 1) to get (6x^2 - 6x).9. Subtracted to get (8x - 5).10. Divided (8x) by (x) to get (8).11. Multiplied (8) by (x - 1) to get (8x - 8).12. Subtracted to get (3).So, the quotient is (8x^3 + 12x^2 + 6x + 8) with a remainder of (3). Therefore, the initial answer provided was incorrect. The correct quotient is (8x^3 + 12x^2 + 6x + 8) with a remainder of (3).Wait, but in the initial problem, the user provided an answer with a quotient of (8x^3 + 12x^2 - 18x + 20) and a remainder of (-25). That suggests that perhaps the initial setup was wrong or there was a miscalculation in the subtraction steps.Let me try to see where the initial answer might have gone wrong. If I follow the initial steps:1. Divided (8x^4) by (x) to get (8x^3).2. Multiplied (8x^3) by (x - 1) to get (8x^4 - 8x^3).3. Subtracted to get (12x^3 - 6x^2 + 2x - 5).4. Divided (12x^3) by (x) to get (12x^2).5. Multiplied (12x^2) by (x - 1) to get (12x^3 - 12x^2).6. Subtracted to get (6x^2 + 2x - 5).7. Divided (6x^2) by (x) to get (6x).8. Multiplied (6x) by (x - 1) to get (6x^2 - 6x).9. Subtracted to get (8x - 5).10. Divided (8x) by (x) to get (8).11. Multiplied (8) by (x - 1) to get (8x - 8).12. Subtracted to get (3).So, the quotient is (8x^3 + 12x^2 + 6x + 8) with a remainder of (3). Therefore, the initial answer provided was incorrect. The correct quotient is (8x^3 + 12x^2 + 6x + 8) with a remainder of (3).Alternatively, perhaps the initial problem was to divide by (x + 1) instead of (x - 1), which would change the signs. Let me check that.If I divide by (x + 1), using synthetic division with (x = -1):-1 | 8 4 -6 2 -5 -8 4 2 -4 ------------------------- 8 -4 -2 4 -9So, the quotient would be (8x^3 - 4x^2 - 2x + 4) with a remainder of (-9). That still doesn't match the initial answer.Alternatively, maybe the initial problem had a different divisor. But as per the user's question, it's (x - 1).Wait, perhaps I made a mistake in the initial long division. Let me try again step by step.Dividing (8x^4 + 4x^3 - 6x^2 + 2x - 5) by (x - 1):1. Divide (8x^4) by (x) to get (8x^3).2. Multiply (8x^3) by (x - 1): (8x^4 - 8x^3).3. Subtract: ( (8x^4 + 4x^3) - (8x^4 - 8x^3) = 12x^3).4. Bring down (-6x^2).5. Divide (12x^3) by (x) to get (12x^2).6. Multiply (12x^2) by (x - 1): (12x^3 - 12x^2).7. Subtract: ( (12x^3 - 6x^2) - (12x^3 - 12x^2) = 6x^2).8. Bring down (2x).9. Divide (6x^2) by (x) to get (6x).10. Multiply (6x) by (x - 1): (6x^2 - 6x).11. Subtract: ( (6x^2 + 2x) - (6x^2 - 6x) = 8x).12. Bring down (-5).13. Divide (8x) by (x) to get (8).14. Multiply (8) by (x - 1): (8x - 8).15. Subtract: ( (8x - 5) - (8x - 8) = 3).So, the quotient is (8x^3 + 12x^2 + 6x + 8) with a remainder of (3). This confirms the earlier result.Therefore, the initial answer provided was incorrect. The correct quotient is (8x^3 + 12x^2 + 6x + 8) with a remainder of (3).Alternatively, perhaps the initial answer was for a different divisor, such as (x + 1) or a different polynomial. But based on the given problem, dividing by (x - 1) yields the quotient (8x^3 + 12x^2 + 6x + 8) with a remainder of (3).In conclusion, after carefully performing the polynomial long division and verifying with synthetic division, the correct quotient is (8x^3 + 12x^2 + 6x + 8) with a remainder of (3).

💡First, I need to determine the number of VIP tickets and general admission tickets sold. I'll start by setting up two equations based on the given information.The first equation represents the total number of tickets sold:V + G = 320The second equation represents the total cost of the tickets:45V + 20G = 7500I'll solve the first equation for V:V = 320 - GNext, I'll substitute this expression for V into the second equation:45(320 - G) + 20G = 7500Expanding and simplifying the equation:14400 - 45G + 20G = 7500-25G = -6900G = 276Now that I have the value for G, I'll find V:V = 320 - 276V = 44Finally, I'll calculate how many fewer VIP tickets were sold than general admission tickets:276 - 44 = 232Therefore, 232 fewer VIP tickets were sold than general admission tickets.

💡Okay, so I have this trigonometry problem here. It says: Given that sin(π/2 + θ) = -1/2, find the value of 2 sin²(θ/2) - 1. The options are A: 1/2, B: -1/2, C: √3/2, and D: ±√3/2. Hmm, let me try to figure this out step by step.First, I remember that sin(π/2 + θ) is a trigonometric identity. I think it relates to cosine because of the co-function identity. Let me recall: sin(π/2 - θ) is equal to cos θ, but here it's sin(π/2 + θ). Maybe it's similar but with a sign change? Let me think. I believe sin(π/2 + θ) is equal to cos θ, but wait, actually, no. Let me check that.Wait, no, sin(π/2 + θ) is actually equal to cos θ, but I think it might have a negative sign because of the angle addition. Let me verify. The sine of (A + B) is sin A cos B + cos A sin B. So, if I apply that to sin(π/2 + θ), it would be sin(π/2)cos θ + cos(π/2)sin θ. I know that sin(π/2) is 1 and cos(π/2) is 0. So, this simplifies to 1 * cos θ + 0 * sin θ, which is just cos θ. So, sin(π/2 + θ) = cos θ. Okay, that makes sense.So, the given equation is sin(π/2 + θ) = -1/2, which translates to cos θ = -1/2. Got it. So, cos θ is -1/2. Now, I need to find 2 sin²(θ/2) - 1. Hmm, that expression looks familiar. I think it's related to the double-angle formula or maybe the half-angle identity.Let me recall the double-angle identities. I remember that cos(2α) = 1 - 2 sin² α. So, if I rearrange that, 2 sin² α = 1 - cos(2α). Therefore, 2 sin²(α) - 1 = -cos(2α). Wait, so if I let α = θ/2, then 2 sin²(θ/2) - 1 = -cos(θ). Is that right? Let me check.Yes, substituting α with θ/2, we get 2 sin²(θ/2) - 1 = -cos θ. So, that means 2 sin²(θ/2) - 1 is equal to negative cosine theta. But from earlier, we found that cos θ is -1/2. So, substituting that in, we get 2 sin²(θ/2) - 1 = -(-1/2) = 1/2. So, the value is 1/2.Wait, let me make sure I didn't make any mistakes here. So, starting from sin(π/2 + θ) = -1/2, which is cos θ = -1/2. Then, using the identity 2 sin²(θ/2) - 1 = -cos θ, so substituting cos θ = -1/2, we get 2 sin²(θ/2) - 1 = -(-1/2) = 1/2. That seems correct.But just to double-check, maybe I can approach this from another angle. Let's see. Since cos θ = -1/2, what are the possible values of θ? Cos θ is -1/2, so θ must be in either the second or third quadrant because cosine is negative there. The reference angle for which cos is 1/2 is π/3, so θ could be 2π/3 or 4π/3. Let me verify.Yes, cos(2π/3) = cos(π - π/3) = -cos(π/3) = -1/2, and cos(4π/3) = cos(π + π/3) = -cos(π/3) = -1/2. So, θ can be 2π/3 + 2πk or 4π/3 + 2πk, where k is any integer.Now, let's compute 2 sin²(θ/2) - 1 for θ = 2π/3. So, θ/2 = π/3. Then, sin(π/3) = √3/2. So, sin²(π/3) = 3/4. Then, 2*(3/4) = 3/2. Subtracting 1, we get 3/2 - 1 = 1/2. Similarly, for θ = 4π/3, θ/2 = 2π/3. Then, sin(2π/3) = √3/2. So, sin²(2π/3) = 3/4. Then, 2*(3/4) = 3/2. Subtracting 1, we get 1/2 again. So, in both cases, the value is 1/2. That confirms my earlier result.Alternatively, I can think about the expression 2 sin²(θ/2) - 1. I remember that this is equivalent to -cos θ, as I derived earlier. So, if cos θ is -1/2, then -cos θ is 1/2. So, that's another way to see it.Just to make sure I didn't confuse any identities, let me recall the half-angle identities. The half-angle identity for sine is sin(θ/2) = ±√[(1 - cos θ)/2]. But in this case, we have 2 sin²(θ/2) - 1, which is a different expression. As I saw earlier, it's related to the double-angle identity for cosine.Wait, another thought: 2 sin²(θ/2) - 1 is equal to -cos θ, which is the same as cos(π - θ). Is that correct? Let me check. Cos(π - θ) is equal to -cos θ, yes. So, 2 sin²(θ/2) - 1 = cos(π - θ). But in this case, since we already know cos θ = -1/2, it's simpler to substitute directly.I think I've covered all the angles here. I've used the co-function identity to find cos θ, then used the double-angle identity to express 2 sin²(θ/2) - 1 in terms of cos θ, substituted the known value, and verified it with specific angle values. All approaches lead me to the same answer, which is 1/2. So, I feel confident that the correct answer is A: 1/2.**Final Answer**boxed{A}